Distance critical

When a jet is supplied at some distance from the surface, the attachment occurs when the distance between the outlet and the surface is below a critical distance otherwise the jet will propagate as a free jet. If the jet attaches to the surface, the flow downstream of the attached point is similar to that of a wall jet. For a compact isothermal jet, the critical distance for jet attachment to the surface is = 6Aq For < 6Al - the velocity decay coefficient Kj... 

The jet will separate from the surface if the axial distance between the slot and the obstruction, x, is less than a specified critical distance x,. (Fig. 7,41). The... 

Obstruction does not affect the jet if the obstruction is further than about eight critical distances (x ) from the slot. In this case the velocity decay of the jet may be obtained from... 

FIGURE 7.41 Critical distance from the slot to the beam fa = width of two dimensional slot, d beam height, x jet core zone length. Reproduced from Holmes and Sachariewicz. 

Graphical interpretation of the factors influencing the critical distance from air supply to the linear obstacle with a height for air supply through a slot diffuser with height Ioq and for air supply through a round nozzle with outlet diameter are presented in Fig. 7.43. Nonisothermal flow has an influence on... 

When ions with this structure diffuse to the vicinity of a negative cathode, the ion is distorted by the field to become polarised, with the positive silver core nearest the cathode (see Fig. 12.2). Once within a critical distance the... 

As symmetrical molecules approach within a critical distance, crystals begin to form in... 

Consider the bimolecular reaction of A and B. The concentration of B is depleted near the still-unreacted A by virtue of the very rapid reaction. This creates a concentration gradient. We shall assume that the reaction occurs at a critical distance tab- At distances r tab. [B] = 0. Beyond this distance, at r > rAB, [B] = [B]°, the bulk concentration of B at r = °°. We shall examine a simplified, two-dimensional derivation the solution in three dimensions must incorporate the mutual diffusion of A and B, requiring vector calculus, and is not presented here. 

Putting the constant equal to zero, implies that <5 = 0 when x = 0, that is that the turbulent boundary layer extends to the leading edge of the surface. An error is introduced by this assumption, but it is found to be small except where the surface is only slightly longer than the critical distance xc for the laminar-turbulent transition. 

Thus, we understand that, in the FSGO model, for some critical distance, the single Gaussian will jump from its symmetric position at the middle of the H-H bond to a dissymetric one represented below. Thus, the FSGO dissociation scheme corresponds to one electron pair on one of the proton (H )and no electron on the second proton (H+) H2 H- + H+... 

For ion-pair formation the electrostatic attraction energy = N iZ j(Q°yi4neQer (per mole of ion pairs) should be larger than the ion pair s mean thermal energy (i.e., at least 2RT). This condition yields for the critical distance of ion pair formation in aqueous solutions at 25°C ... 

UNEQ can be applied when only a few variables must be considered. It is based on the Mahalanobis distance from the centroid of the class. When this distance exceeds a critical distance, the object is an outlier and therefore not part of the class. Since for each class one uses its own covariance matrix, it is somewhat related to QDA (Section 33.2.3). The situation described here is very similar to that discussed for multivariate quality control in Chapter 20. In eq. (20.10) the original variables are used. This equation can therefore also be used for UNEQ. For convenience it is repeated here. 

A useful tool in the interpretation of SIMCA is the so-called Coomans plot [32]. It is applied to the discrimination of two classes (Fig. 33.18). The distance from the model for class 1 is plotted against that from model 2. On both axes, one indicates the critical distances. In this way, one defines four zones class 1, class 2, overlap of class 1 and 2 and neither class 1 nor class 2. By plotting objects in this plot, their classification is immediately clear. It is also easy to visualize how certain a classification is. In Fig. 33.18, object a is very clearly within class 1, object b is on the border of that class but is not close to class 2 and object c clearly belongs to neither class. 

If neighboring roots are separated by more than the critical distance, then inlerroot competition can be neglected. 

It can be shown (see Southwell, 1913) that the critical distance between stiffeners, Lc, beyond which stiffening will not be effective is given by ... 

J. Bjerrum (1926) first developed the theory of ion association. He introduced the concept of a certain critical distance between the cation and the anion at which the electrostatic attractive force is balanced by the mean force corresponding to thermal motion. The energy of the ion is at a minimum at this distance. The method of calculation is analogous to that of Debye and Hiickel in the theory of activity coefficients (see Section 1.3.1). The probability Pt dr has to be found for the ith ion species to be present in a volume element in the shape of a spherical shell with thickness dr at a sufficiently small distance r from the central ion (index k). 

Table 6.4. Typical Values for the Critical Distance Ro Associated with Long-Range...

However, if the molecular configuration is such as to bring the radical centers into close contact (say, to within some critical distance D), interaction will begin and the two doublet states will collapse since the spins of the two electrons are no longer independent of each other. Quantum mechanics tells us that under these conditions the total number of states is dependent upon the sum of the spin quantum numbers for the two electrons ... 

Thus the theoretical critical distance can be calculated entirely on the basis of spectroscopic properties of the donor and acceptor. 

All of the examples of singlet energy transfer we have considered take place via the long-range resonance mechanism. When the oscillator strength of the acceptor is very small (for example, n-> n transitions) so that the Fdrster critical distance R0 approaches or is less than the collision diameter of the donor-acceptor pair, then all evidence indicates that the transfer takes place at a diffusion-controlled rate. Consequently, the transfer mechanism should involve exchange as well as Coulomb interaction. Good examples of this type of transfer have been provided by Dubois and co-workers.(47-49)... 

If the critical separation is determined for a large number of relative geometries of the electron and molecule it is possible to obtain a three-dimensional picture of the probability of ionization as a function of the orientation of the molecule. Effectively, the idea of an ionization cross section, the area the target molecule presents to the electron, is extended to a three-dimensional object defined by the critical distances, with ionization occurring when the electron penetrates the surface enclosing this volume. The volume enclosed by the electron impact ionization surface may be used to obtain an estimate for the cross section (volume averaged cross section) ... 

FIGURE 4.5 Clustering in ID separations, (a) Resolution between adjacent SCs defines the critical distance Axo. (b) interdistances between adjacent SCs (fourth through seventh SC) are considered. I > Ax0, II < Ax0, and III > Ax0. Thus fifth and sixth SC are merged in the same band (doublet) that is completely separated from the previous and subsequent bands. 

FIGURE 4.6 Scheme of the SMO procedure to estimate the number of SC. (a) Original line chromatogram, (b, c) Multiplets are formed by applying increasing critical distance values, Ax 0 ... 

F4FlAsH-P12 61 at 544 nm expanded the spectral range of the biarsenical dyes. Furthermore, the fluorescence lifetime increased to a value of 5.2 ns. Owing to the results of this study, the authors suggested that the two compounds (57 and 58) would form an excellent FRET pair with a large critical distance. In addition, they envisage that the observed characteristics should facilitate improved structural and dynamic studies of proteins in living cells. 

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